Mathematical Methods for Physics and Engineering - Matematica.NET

30.2 PROBABILITYA 4A 3OA 1A 2BFigure 30.5A collection of traffic islands connected by one-way roads.i =1, 2, 3, 4. From figure 30.5, we see also thatPr(B|A 1 )= 1 , Pr(B|A 3 2)= 1 , Pr(B|A 3 3)=0, Pr(B|A 4 )= 2 = 1 . 4 2Thus, using the total probability law (30.24), we find that the probability of arriving at Bis given byPr(B) = ∑ (Pr(A i )Pr(B|A i )= 1 1 + 1 +0+ ) 14 3 3 2 =7. ◭ 24iFinally, we note that the concept of conditional probability may be straightforwardlyextended to several compound events. For example, in the case of threeevents A, B, C, we may write Pr(A ∩ B ∩ C) in several ways, e.g.Pr(A ∩ B ∩ C) =Pr(C)Pr(A ∩ B|C)=Pr(B ∩ C)Pr(A|B ∩ C)=Pr(C)Pr(B|C)Pr(A|B ∩ C).◮Suppose {A i } is a set of mutually exclusive events that exhausts the sample space S. IfBand C are two other events in S, show thatPr(B|C) = ∑ iPr(A i |C)Pr(B|A i ∩ C).Using (30.19) and (30.17), we may writePr(C)Pr(B|C) =Pr(B ∩ C) = ∑ iPr(A i ∩ B ∩ C). (30.25)Each term in the sum on the RHS can be expanded as an appropriate product ofconditional probabilities,Pr(A i ∩ B ∩ C) =Pr(C)Pr(A i |C)Pr(B|A i ∩ C).Substituting this form into (30.25) and dividing through by Pr(C) gives the requiredresult. ◭1131